Final answer:
To prove the trigonometric identity cos6 x + sin6 x = 1 - ¾ sin2 2x, we start with the left-hand side, expand it using the binomial theorem, and apply trigonometric identities such as cos2x = 1 - sin2x. We simplify the expression and share the result with the right-hand side, using the double-angle formula.
Step-by-step explanation:
To prove that cos6 x + sin6 x = 1 - ¾ sin2 2x, let's start by expanding the left-hand side using the binomial theorem and known trigonometric identities.
- First, we express cos6x and sin6x as (cos2x)3 and (sin2x)3, respectively.
- Next, we use the identity cos2x = 1 - sin2x to rewrite cos6x.
- Similarly, we use sin2x = 1 - cos2x to rewrite sin6x.
- Now, we have two expressions in terms of sin2x and cos2x, which we can further simplify.
- Combining the two expressions and simplifying, we arrive at cos6x + sin6x in terms of sin2x and cos2x.
- To introduce sin2 2x into our expression, we use the double-angle formula sin 2x = 2 sin x cos x.
- After some algebraic manipulation, we end up with 1 - ¾ sin2 2x, completing the proof.